ELECTROMAGNETIC FIELD ANALYSIS 499 



response to the general electromotive force is expressed in terms of responses 

 to the unit step function, or the unit impulse function, or the steady state 

 responses at various frequencies. 



There are numerous variations of the same general idea, some of which 

 are more suitable to one class of problems and others to another class/ 

 If the distribution of electric charge and current is known, then in many 

 cases (but not in all) it is best to subdivide it into small volume elements. 

 Except for a possible static electric charge distribution, the elements will 

 be dipoles. The entire field can thus be regarded as the resultant of spherical 

 waves generated by dipoles of given moment and position. To simplify 

 the integration involved in this method certain auxiliary functions, called 

 the retarded potentials, are introduced. One should not try to ascribe 

 to these auxiliary mathematical functions any physical significance and one 

 should always remember that on certain occasions potential functions, 

 other than the retarded potentials, turn out to be more useful. We should 

 also keep in mind that, in order to apply this method, we have to know the 

 complete distribution of electric conduction currents and as a general rule 

 we do not have this information. Consider, for instance, the problem of 

 electromagnetic shielding. The current in the coil is given; but that in 

 the shield has to be determined. There are methods for calculating the 

 induced current; but these methods give at the same time the shielding 

 effectiveness, and that without employing retarded potentials. It is in 

 approximate studies of radiation patterns of antennas and antenna arrays 

 that the retarded potential method is displayed to the best advantage. 



The retarded potentials are based on representation of fields in terms of 

 spherical coordinates; that is, in terms of fields associated with hypothetical 

 point sources at the origin of the coordinate system. General fields can 

 also be expressed in terms of cylindrical coordinates and, consequently, in 

 terms of fields associated with hypothetical line sources situated along the 

 axis of the coordinate system. Likewise, fields can be expressed in cartesian 

 coordinates; that is, in terms of "plane waves". All such representations 

 have useful applications. The current in the coil is given. 



Discontinuities 



In the analysis of the various transmission modes for a given wave guide 

 it is assumed at first that the boundaries of the wave guide are analytic 

 functions of the coordinates. Any discontinuity or irregularity has to be 

 treated separately, simply because there is nothing in the analytic part 

 of the wave guide to suggest that a discontinuity might occur, or to prescribe 

 the properties of this discontinuity. Discontinuities may be accidental, 

 unavoidable or intentional. A kink in a wire is an example of an accidental 



